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On some model problem for the propagation of interacting species in a special environment
Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues
1. | Ingenium College of Liberal Arts, Kwangwoon University, Seoul 01891, Korea |
2. | Division of Medical Mathematics, National Institute for Mathematical Sciences, Daejeon 34047, Korea |
3. | Department of Mathematics, Kyung Hee University, Seoul 02447, Korea |
4. | School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Korea |
5. | Samsung Fire & Marine Insurance, Seoul 04523, Korea |
6. | Department of Undergraduate Studies, Daegu Gyeongbuk Institute of Science and Technology, Daegu 42988, Korea |
In this paper, we study the dynamic phase transition for one dimensional Brusselator model. By the linear stability analysis, we define two critical numbers $ {\lambda}_0 $ and $ {\lambda}_1 $ for the control parameter $ {\lambda} $ in the equation. Motivated by [
References:
[1] |
R. Anguelov and S. M. Stoltz,
Stationary and oscillatory patterns in a coupled Brusselator model, Math. Computers Simul., 133 (2017), 39-46.
doi: 10.1016/j.matcom.2015.06.002. |
[2] |
K. J. Brown and F. A. Davidson,
Global bifurcation in the Brusselator system, Nonlin. Anal., 12 (1995), 1713-1725.
doi: 10.1016/0362-546X(94)00218-7. |
[3] | I. R. Epstein and J. A. Pojman, An Introduction to Nonlinear Chemical Dynamics,, Oxford Univ. Press, 1998. Google Scholar |
[4] |
M. Ghergu,
Non-constant steady-state solutions for Brusselator type systems, Nonlinearity, 21 (2008), 2331-2345.
doi: 10.1088/0951-7715/21/10/007. |
[5] |
B. Guo and Y. Han,
Attractor and spatial chaos for the Brusselator in $ \mathbb{R}^N$, Nonlin. Anal., 70 (2009), 3917-3931.
doi: 10.1016/j.na.2008.08.002. |
[6] |
H. Kang and Y. Pesin,
Dynamics of a discrete brusselator model: Escape to infinity and julia set, Milan J. Math., 73 (2005), 1-17.
doi: 10.1007/s00032-005-0036-y. |
[7] |
H. Shoji, K. Yamada, D. Ueyama and T. Ohta, Turing patterns in three dimensions, Phys. Rev. E, 75 (2007), 046212, 13 pp.
doi: 10.1103/PhysRevE.75.046212. |
[8] |
T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific, 2005.
doi: 10.1142/9789812701152. |
[9] |
T. Ma and S. Wang, Phase transitions for the Brusselator model, J. Math. Phys., 52 (2011), 033501, 23 pp.
doi: 10.1063/1.3559120. |
[10] |
T. Ma and S. Wang, Phase Transition Dynamics 2nd ed., Springer, 2019.
doi: 10.1007/978-3-030-29260-7. |
[11] |
MathWorks, Matlab: Mathematics(R2020a), Retrieved from https://www.mathworks.com/help/pdf_doc/matlab_math.pdf Google Scholar |
[12] |
A. S. Mikhailov and K. Showalter,
Control of waves, patterns and turbulence in chemical systems, Physics Reports, 425 (2006), 79-194.
doi: 10.1016/j.physrep.2005.11.003. |
[13] |
L. A. Peletier and W. C. Troy, Spatial Patterns: Higher Order Models in Physics and Mechanics, Birkhauser, 2001.
doi: 10.1007/978-1-4612-0135-9. |
[14] |
R. Peng and M. X. Wang,
Pattern formation in the Brusselator system, J. Math. Anal. Appl., 309 (2005), 151-166.
doi: 10.1016/j.jmaa.2004.12.026. |
[15] |
R. Peng and M. X. Wang,
On steady-state solutions of the Brusselator-type system, Nonlin. Anal., 71 (2009), 1389-1394.
doi: 10.1016/j.na.2008.12.003. |
[16] |
I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems, J. Chem. Phys., 48 (1968), 1695-1700. Google Scholar |
[17] |
A. Toth, V. Gaspar and K. Showalter,
Signal transmission in chemical systems: Propagation of chemical waves through capillary tubes, J. Phys. Chem., 98 (1994), 522-531.
doi: 10.1021/j100053a029. |
[18] |
A. M. Turing,
The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
[19] |
Y. You,
Global Dynamics of the Brusselator equations, Dynamics of PDE, 4 (2007), 167-196.
doi: 10.4310/DPDE.2007.v4.n2.a4. |
show all references
References:
[1] |
R. Anguelov and S. M. Stoltz,
Stationary and oscillatory patterns in a coupled Brusselator model, Math. Computers Simul., 133 (2017), 39-46.
doi: 10.1016/j.matcom.2015.06.002. |
[2] |
K. J. Brown and F. A. Davidson,
Global bifurcation in the Brusselator system, Nonlin. Anal., 12 (1995), 1713-1725.
doi: 10.1016/0362-546X(94)00218-7. |
[3] | I. R. Epstein and J. A. Pojman, An Introduction to Nonlinear Chemical Dynamics,, Oxford Univ. Press, 1998. Google Scholar |
[4] |
M. Ghergu,
Non-constant steady-state solutions for Brusselator type systems, Nonlinearity, 21 (2008), 2331-2345.
doi: 10.1088/0951-7715/21/10/007. |
[5] |
B. Guo and Y. Han,
Attractor and spatial chaos for the Brusselator in $ \mathbb{R}^N$, Nonlin. Anal., 70 (2009), 3917-3931.
doi: 10.1016/j.na.2008.08.002. |
[6] |
H. Kang and Y. Pesin,
Dynamics of a discrete brusselator model: Escape to infinity and julia set, Milan J. Math., 73 (2005), 1-17.
doi: 10.1007/s00032-005-0036-y. |
[7] |
H. Shoji, K. Yamada, D. Ueyama and T. Ohta, Turing patterns in three dimensions, Phys. Rev. E, 75 (2007), 046212, 13 pp.
doi: 10.1103/PhysRevE.75.046212. |
[8] |
T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific, 2005.
doi: 10.1142/9789812701152. |
[9] |
T. Ma and S. Wang, Phase transitions for the Brusselator model, J. Math. Phys., 52 (2011), 033501, 23 pp.
doi: 10.1063/1.3559120. |
[10] |
T. Ma and S. Wang, Phase Transition Dynamics 2nd ed., Springer, 2019.
doi: 10.1007/978-3-030-29260-7. |
[11] |
MathWorks, Matlab: Mathematics(R2020a), Retrieved from https://www.mathworks.com/help/pdf_doc/matlab_math.pdf Google Scholar |
[12] |
A. S. Mikhailov and K. Showalter,
Control of waves, patterns and turbulence in chemical systems, Physics Reports, 425 (2006), 79-194.
doi: 10.1016/j.physrep.2005.11.003. |
[13] |
L. A. Peletier and W. C. Troy, Spatial Patterns: Higher Order Models in Physics and Mechanics, Birkhauser, 2001.
doi: 10.1007/978-1-4612-0135-9. |
[14] |
R. Peng and M. X. Wang,
Pattern formation in the Brusselator system, J. Math. Anal. Appl., 309 (2005), 151-166.
doi: 10.1016/j.jmaa.2004.12.026. |
[15] |
R. Peng and M. X. Wang,
On steady-state solutions of the Brusselator-type system, Nonlin. Anal., 71 (2009), 1389-1394.
doi: 10.1016/j.na.2008.12.003. |
[16] |
I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems, J. Chem. Phys., 48 (1968), 1695-1700. Google Scholar |
[17] |
A. Toth, V. Gaspar and K. Showalter,
Signal transmission in chemical systems: Propagation of chemical waves through capillary tubes, J. Phys. Chem., 98 (1994), 522-531.
doi: 10.1021/j100053a029. |
[18] |
A. M. Turing,
The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
[19] |
Y. You,
Global Dynamics of the Brusselator equations, Dynamics of PDE, 4 (2007), 167-196.
doi: 10.4310/DPDE.2007.v4.n2.a4. |








subcases | ||||
(ⅰ-1) | stable | saddle | ||
(ⅰ-2) | saddle | stable | ||
(ⅰ-3) | stable | saddle | ||
(ⅰ-4) | saddle | stable |
subcases | ||||
(ⅰ-1) | stable | saddle | ||
(ⅰ-2) | saddle | stable | ||
(ⅰ-3) | stable | saddle | ||
(ⅰ-4) | saddle | stable |
subcases | ||||
(ⅱ-1) | stable | saddle | ||
(ⅱ-2) | saddle | stable | ||
(ⅱ-3) | stable | saddle | ||
(ⅱ-4) | saddle | stable |
subcases | ||||
(ⅱ-1) | stable | saddle | ||
(ⅱ-2) | saddle | stable | ||
(ⅱ-3) | stable | saddle | ||
(ⅱ-4) | saddle | stable |
subcases | ||||||
(ⅲ-1) | stable | saddle | saddle | stable | ||
(ⅲ-2) | saddle | stable | stable | saddle | ||
(ⅲ-3) | stable | saddle | saddle | stable | ||
(ⅲ-4) | saddle | stable | stable | saddle |
subcases | ||||||
(ⅲ-1) | stable | saddle | saddle | stable | ||
(ⅲ-2) | saddle | stable | stable | saddle | ||
(ⅲ-3) | stable | saddle | saddle | stable | ||
(ⅲ-4) | saddle | stable | stable | saddle |
subcases | ||||
(ⅳ-1) | stable | stable | saddle | saddle |
(ⅳ-2) | saddle | saddle | stable | stable |
subcases | ||||
(ⅳ-1) | stable | stable | saddle | saddle |
(ⅳ-2) | saddle | saddle | stable | stable |
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